Adaptive joint fuzzy sets for function approximation

This paper presents a new method to create and tune joint fuzzy sets. Multidimensional fuzzy sets define the if-part fuzzy sets of rules in feedforward fuzzy function approximators. These joint set functions do not factor into a product of scalar fuzzy sets (such as triangles or bell curves) and so they do not ignore the correlation structure among the input components. The joint set functions transform a scalar distance measure that preserves the correlation structure. Supervised learning tunes the metrical joint set functions and tunes the scalar set functions that make up factorable joint set functions. Factorable joint set functions tend to collapse to spikes in high dimensions. This holds for all joint set functions that combine factors with product or minimum or other t-norms. Simulations suggest that some metrical joint set functions may offer a practical tool for fuzzy function approximation in higher dimensions and in L^p function spaces